What is the integral of (e^x(xln(x)+1))/x ?

The integral of (e^x(xln(x)+1))/x is e^xln(x)+C

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integral of (e^x(xln(x)+1))/x

\int \frac{e ^{x} ( x ln ( x ) + 1 )}{x} d x

e^{x} ln (x) + C

Solution steps

\int \frac{e ^{x} ( x ln ( x ) + 1 )}{x} d x

Expand

\frac{e ^{x} ( x ln ( x ) + 1 )}{x} : e^{x} ln (x) + \frac{e ^{x}}{x}

= \int e^{x} ln (x) + \frac{e ^{x}}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{x} ln (x) d x + \int \frac{e ^{x}}{x} d x

\int e^{x} ln (x) d x = e^{x} ln (x) - Ei (x)

\int \frac{e ^{x}}{x} d x = Ei (x)

= e^{x} ln (x) - Ei (x) + Ei (x)

Simplify

= e^{x} ln (x)

Add a constant to the solution

= e^{x} ln (x) + C

\frac{3 x + 8}{x ^{2}}

\frac{( 3 s + 1 )}{s + 2}

\frac{\partial}{\partial y} (\frac{x}{( x + y ) ^{2}})

\int_{1}^{3} (x^{3} + \frac{1}{4 x ^{3}}) d x

y = 2 x - 1 + 4 x^{2}, (1, 5)

What is the integral of (e^x(xln(x)+1))/x ?
The integral of (e^x(xln(x)+1))/x is e^xln(x)+C